#8107640

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#2431	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4396 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

(1,2)(3,8)(4,6)(5,7)(9,10,15,12)(11,16,17,13)(14,20,22,18)(19,21,24,23)(25,26)(27,33)(28,32)(29,30)(31,35)(34,39)(36,37)(38,40);(1,3,6,5)(2,7,4,8)(9,11,15,17)(10,16,12,13)(14,19,22,24)(18,23,20,21)(26,30)(31,33)(32,37)(38,39);(1,3,6,5)(2,8,4,7)(10,12)(11,17)(18,20)(19,24)(25,26)(27,31)(28,32)(29,30)(33,35)(34,38)(36,37)(39,40);(1,4,6,2)(3,8,5,7)(9,12,15,10)(11,16,17,13)(14,20,22,18)(19,23,24,21)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,40)(37,39);(9,13,15,16)(10,17,12,11)(14,21,22,23)(18,24,20,19);(1,5,6,3)(2,8,4,7)(9,14)(10,18)(11,19)(12,20)(13,21)(15,22)(16,23)(17,24)(25,28,29,36)(26,32,30,37)(27,34,35,40)(31,38,33,39)

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=ghi,d^2=gi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8014836);